Question: To play basketball with her friends, Evangeline needs to pump air in her ball, which is completely deflated. Before inflating it, the ball weighs $0.615$ kilograms. Afterwards, it weighs $0.624$ kilograms. The diameter of the ball is $0.24$ meters. Assuming the inflated ball is perfectly spherical, what is the air density within it? Round your answer, if necessary, to the nearest hundredth.
Explanation: This is a density word problem. To solve it, we can use the following equation, which is the volume definition of density: ${\text{Density}}=\dfrac{{\text{Total quantity}}}{{\text{Volume}}}$ What do we know? The ball's diameter is $0.24$ meters (we can use this to find the ${\text{volume}}$ ). When completely deflated, the ball weighs $0.615$ kilograms, and when inflated, it weighs $0.624$ kilograms. The difference gives us the weight of the air, which is the ${\text{total quantity}}$. What do we need to find? The ${\text{density}}$ of the air inside the ball. The ${\text{total quantity}}$ of the air is $0.624-0.615={0.009}$ kilograms. The radius of the ball is $\dfrac{0.24}{2}=0.12$ meters. So the ${\text{volume}}$ of the ball is ${\dfrac43\pi\cdot 0.12^3}$ cubic meters (we will wait with evaluating this expression until the end of the solution). Now we can plug ${\text{total quantity}=0.009}$ and ${\text{volume}=\dfrac43\pi\cdot 0.12^3}$ in the equation. $\begin{aligned} {\text{Density}}&=\dfrac{{\text{Total quantity}}}{{\text{Volume}}} \\\\ &=\dfrac{{0.009}}{{\dfrac43\pi\cdot 0.12^3}} \\\\ &\approx 1.24 \end{aligned}$ The air density in the inflated ball is approximately $1.24$ kilograms per cubic meter.